EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
November 3, 2011
Problem Set 6
Due Date: 11/11/11, 11:11:11pm
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Instructions: Solutions should be put in my mailbox, slipped under the door of my office,
or emailed to [email protected]. Starred problems are harder and should be a lower
priority.
1. Investment with Adjustment Costs
Another “canonical” model that often gets analyzed using the maximum principle is
the one solved by a firm that faces costs of adjusting its capital stock. During each
period t ∈ [0, ∞), this firm produces a flow of Y (t) units of output using a stock of
K(t) units of capital according to the production function
Y (t) = K(t)α ,
with 0 < α < 1. The capital stock depreciates at the constant rate δ, where 0 < δ < 1;
hence, by investing I(t) units of output at time t, the firm augments its capital stock
according to
K̇(t) = I(t) − δK(t)
At the same time, however, the firm incurs a quadratic cost of adjustment given by
φ
I(t)2 ,
2
where the parameter φ > 0 governs the magnitude of this cost. In this problem, the
firm’s choice of I(t) can be positive, in which case the firm is installing new capital,
or negative, in which case the firm is selling off existing capital; but either way, the
formulation implies that it will incur adjustment costs in making these changes. The
firm sells off whatever output remains after its investment choice is made; its profits
during period t are therefore
K(t)α − I(t) −
φ
I(t)2 .
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Finally, let r > 0 denote the constant discount rate, reflecting either impatience on the
part of the firm’s owners or a positive rate of interest at which profits received today
can generate additional income if saved for the future.
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November 11, 2011, 11:11pm
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The firm’s problem can now be stated as: choose continuously differentiable functions
I(t) and K(t) for t ∈ [0, ∞) to maximize the discounted value of profits over the infinite
horizon,
Z ∞
φ
e−rt [K(t)α − I(t) − I(t)2 ] dt,
2
0
subject to the capital accumulation equation for all t ∈ [0, ∞), taking the initial capital
stock K(0) − K̄0 > 0 as given.
(a) Define the Hamiltonian for this problem and write down the first-order condition
and the pair of differential equations that, according to the maximum principle,
are necessary conditions for the values of I(t) and K(t) that solve the firm’s
infinite-horizon problem.
(b) Use the necessary conditions to rewrite the system as one involving two differential
equations in two unknowns: investment I(t) and the capital stock K(t).
(c) Now write down a set of two equations that determine the optimal steady-state
values I ∗ for investment and K ∗ for the capital stock (Note: it will not be possible
to actually solve these equations for I ∗ and K ∗ algebraically; if you want to
find these steady-state values you would have to assign numerical values to each
of the model’s parameters and then use a calculator or a computer to find the
corresponding numerical values of I ∗ and K ∗ ).
(d) Finally, use your results from above to draw a phase diagram that illustrates
the following property of the solution to the firm’s problem: starting from any
value K(0) > 0 for the initial capital stock, there is a unique value of investment
I(0) such that, starting from I(0) and K(0), the optimally-chosen paths for I(t)
and K(t) converge to the steady-state values I ∗ and K ∗ . In drawing this phase
diagram, it may be helpful to note that while investment I(t) can take on positive
or negative values, depending on whether the firm is accumulating or selling off
capital, the capital stock K(t) itself must always remain positive when these
variables are chosen optimally.
(e) Are the necessary conditions above also sufficient? Explain with mathematical
arguments.
2. Natural Resource Depletion in Discrete Time
Let ct denote society’s consumption of an exhaustible natural resource at each date
t = 0, 1, 2, ..., and suppose that a representative consumer gets utility from this resource
as described by
∞
X
β t ln(ct )
(1)
t=0
where the discount factor satisfies 0 < β < 1. Let st denote the stock of the resource
that remains at the beginning of each period t = 0, 1, 2, .... Since the resource in
nonrenewable, this stock evolves according to the constraint
st+1 = st − ct
2
(2)
which indicates that consumption during period t just subtracts from the stock, and
the stock does not get replenished. Consider a social planner, who chooses sequences
∞
{ct }∞
t=0 and {st }t=1 to maximize the representative consumer’s utility subject to the
stock evolution equation, which must hold for all periods t = 0, 1, 2, ..., taking the
initial stock s0 = s̄0 as given, and that st ≥ 0 at all times.
(a) Write down the current-value Hamiltonian and the necessary conditions a solution
must satisfy according to the Maximum Principle. Be sure to write down the
relevant transversality condition.2
(b) Given the necessary conditions, what can you conclude about the time-path of
consumption. Is it increasing, constant, or decreasing over time, or is it oscillating
over time?
(c) Show that the consumption at time t is given by ct = β t c0 , where c0 is the
consumption at time 0 along the optimum.
(d) Show that the transversality condition implies that limt→∞ st+1 = 0.
(e) * Show that the stock evolution equation implies st+1 = s̄0 −
1−β t+1
c0 .
1−β
(f) Show that c0 = (1 − β)s̄0 , and thus the optimal consumption at time t is ct =
β t (1 − β)s̄0 .
3. Sustainable Fishing
The undisturbed rate of growth of a population of fish in a certain lake is given by
Ṅ (t) = aN (t) − bN 2 (t), where a, b > 0 and N (t) is the population at time t. Let
c(t) denote the rate at which fish are being harvested for consumption by cooperative
community that owns the lake, and u(c) the flow utility from consumption rate c. The
cooperative wishes to sustainably maximize the present value of the discounted stream
of flow utility by optimally choosing a fish harvesting plan, where future utility is discounted at rate ρ > 0. Thus, the cooperative solves the following dynamic optimization
problem:
Z
∞
e−ρt u(c(t))dt
max
c,N
0
subject to
Ṅ (t) = aN (t) − bN 2 (t) − c(t),
a
N (0) = .
b
Assume that u0 > 0 and u00 < 0, and that a > ρ.
(a) Write down the Hamiltonian in current-value form and find the necessary conditions that the optimal consumption c∗ (t) and population N ∗ (t) paths and that the
current-value costate path µ∗ (t) must satisfy, according to the Maximum Principle. What is the necessary transversality condition?
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See Ireland’s 2010 notes about the Maximum Principle, available at the class website, if you are unsure
about how to write the current-value Hamiltonian for discrete time problems.
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(b) Draw a phase diagram for µ∗ and N ∗ . Depict the saddle-point stable path and a
sample trajectory other than the saddle-point stable path.
(c) * Can you prove that the condition for a saddle-point stable path hold?
(d) Suppose that the utility function is of CES form i.e.
u(c) =
c1−σ − 1
,
1−σ
with σ > 0. Simplify the necessary conditions you found above using this assumption about the utility function and draw a phase diagram for c∗ and N ∗ .
(e) Why is it important that N (0) be no greater than ab ? Do you have any intuition
why assuming N (0) = ab makes sense? (Hint: what happens in a world without
fishing?)
(f) How does the steady-state consumption level and fish population level given CES
utility depend upon:
i. the discount rate ρ?
ii. the coefficient of relative risk aversion σ?
(g) * Show that the steady-state values of N ∗ and c∗ you found with CES utility is
exactly the same for any strictly concave utility function. What is the intuition
for why this is true?
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